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Magazine Chapter 2: Problem 69
Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -32 \leq 14(12 x-1)+34<32 $$
Short Answer
Expert verified
\( \left[ \frac{-13}{42}, \frac{1}{14} \right) \)
Step by step solution
01
Expand and Simplify the Inequality
First, distribute the 14 in the expression inside the inequality. Break it down into two separate inequalities:1. \[-32 \leq 14(12x-1) + 34\]Distribute the 14:\[-32 \leq 168x - 14 + 34\] Combine like terms:\[-32 \leq 168x + 20\]2. \[168x - 14 + 34 < 32\]Combine like terms:\[168x + 20 < 32\]
02
Solve the First Inequality
Now, solve the first inequality from Step 1:\[-32 \leq 168x + 20\] Subtract 20 from both sides:\[-52 \leq 168x\] Divide by 168:\[x \geq \frac{-52}{168}\]Simplify the fraction:\[x \geq \frac{-13}{42}\]
03
Solve the Second Inequality
Next, solve the second inequality from Step 1:\[168x + 20 < 32\] Subtract 20 from both sides:\[168x < 12\] Divide by 168:\[x < \frac{12}{168}\]Simplify the fraction:\[x < \frac{1}{14}\]
04
Combine Solution for the Inequality
Combine the solutions from Step 2 and Step 3 to find the range of values for \(x\):\[ \frac{-13}{42} \leq x < \frac{1}{14} \]This represents the solution set for the inequality.
05
Graph the Solution Set
Draw a number line and shade the region that matches \( \frac{-13}{42} \leq x < \frac{1}{14} \). Notice that \(\frac{-13}{42}\) has a closed dot indicating it is included, while \(\frac{1}{14}\) has an open dot indicating it is not included.
06
Write the Solution in Interval Notation
Express the solution set in interval notation:\[ \left[ \frac{-13}{42}, \frac{1}{14} \right) \]This indicates the interval includes \(\frac{-13}{42}\) but does not include \(\frac{1}{14}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way to describe the set of solutions of inequalities using interval brackets. It simplifies the expression of ranges where a variable falls between two numbers. Consider the expression \( \frac{-13}{42} \leq x < \frac{1}{14} \). In interval notation, this can be expressed as \( \left[ \frac{-13}{42}, \frac{1}{14} \right) \).
Here鈥檚 a brief breakdown of the different types of brackets used in interval notation:
Here鈥檚 a brief breakdown of the different types of brackets used in interval notation:
- [ ]: These brackets indicate that the endpoint is included in the interval. For example, \( [a, b] \) means "\( a \)" and "\( b \)" are part of the solution set.
- ( ): These brackets show that the endpoint is not included. Thus, \( (a, b) \) means neither "\( a \)" nor "\( b \)" is included.
Graphical Representation
Visualizing inequalities with a graph provides a straightforward way to understand solution sets. When an inequality is solved, a number line is often used to represent the range of possible values. Here鈥檚 how to graph the solution set \( \frac{-13}{42} \leq x < \frac{1}{14} \):
- Draw a number line where both \( \frac{-13}{42} \) and \( \frac{1}{14} \) are marked as key points.
- Place a closed dot on \( \frac{-13}{42} \). This indicates that \( x \) can equal \( \frac{-13}{42} \) since the inequality is \( \leq \).
- Place an open dot on \( \frac{1}{14} \). This shows that \( x \) cannot equal \( \frac{1}{14} \), as it is just "less than".
Solution Set
A solution set contains all the possible solutions that satisfy a given inequality or equation. For the problem \(-32 \leq 14(12 x-1)+34 <32 \), the solution set comprises values of \( x \) that make this condition true.
After simplifying and solving the inequalities step by step, we found that:
This means any value of \( x \) within this range satisfies the original inequality. Understanding the concept of a solution set helps in solving not just single inequalities but also systems of inequalities, offering a way to evaluate multiple constraints simultaneously.
After simplifying and solving the inequalities step by step, we found that:
- \( x \geq \frac{-13}{42} \) from the first inequality.
- \( x < \frac{1}{14} \) from the second inequality.
This means any value of \( x \) within this range satisfies the original inequality. Understanding the concept of a solution set helps in solving not just single inequalities but also systems of inequalities, offering a way to evaluate multiple constraints simultaneously.
